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W. Auzinger:
"Exponential and rational multistep methods";
Talk: Second joint Conference of Austrian and Mongolian Mathematicians, Ulaanbaatar (invited); 2011-10-21 - 2011-10-22.

Rational multistep methods are a blend of one-step and multistep methods, for the purpose of efficient integration of stiff systems of differential equations, especially after semi-discretization of evolutionary partial differential equations. For a semi-linear stiff system u'(t) = Au(t)+ f(t,u(t)), u(0) = u0, a rational multistep integrator involves an A-stable rational approximation of the (stiff) linear part in combination with a multistep ansatz for the (non-stiff or mildly stiff) nonlinear part.

Since the 1970´s, several authors have proposed such schemes, but a convergence analysis has not been available until recently. It has been demonstrated how rational multistep schemes can be interpreted in a natural way as rational modifications of so-called exponential multistep schemes. This point of view reveals a strategy to analyze convergence and to obtain rigorous error bounds. We explain this approach, discuss its interrelation with exponential schemes, and sketch the convergence proof for the so-called Adams-Padé version, making use of results from approximation theory concerning the error of Padeé approximations to the matrix exponential exp(tA). Furthermore, implementation issues and numerical examples are discussed.